Three charged particles lie along the y-axis, as shown in the image on the right. The charge of each particle is as follows: 20C =2.0 C =-3.0 C 9red 5.0 C 40 cm -3.0C You are told the green particle is at equilibrium. What is the separation distance, in centimeters, between the yellow particle and green particle?
The equilibrium condition for charged particles states that the net force on each particle must be zero. This means that the vector sum of the forces between each pair of particles must add up to zero.
The force between two charged particles can be calculated using Coulomb's law: F = k * q1 * q2 / r^2, where F is the force between the particles, k is the electrostatic constant (9 * 10^9 Nm^2/C^2), q1 and q2 are the charges of the particles, and r is the separation distance between the particles.
Since the green particle is at equilibrium, the vector sum of the forces between the green particle and the other two particles must be zero.
Let's first consider the force between the green and yellow particles. The force between two charged particles is attractive if the charges have opposite signs, and repulsive if the charges have the same sign. In this case, the green particle has a positive charge (+2.0 C) and the yellow particle has a negative charge (-3.0 C), so the force is attractive.
Setting up the equation for the forces between the green and yellow particles:
F_green_yellow = k * q_green * q_yellow / r_green_yellow^2
Since we are told that the green particle is at equilibrium, the force between the green and yellow particles must be equal in magnitude to the force between the green particle and the red particle.
So, we can write:
F_green_yellow = F_green_red
k * q_green * q_yellow / r_green_yellow^2 = k * q_green * q_red / r_green_red^2
q_yellow / r_green_yellow^2 = q_red / r_green_red^2
Since we are asked to find the separation distance between the yellow and green particles, we can set r_green_yellow as x (the unknown distance) and r_green_red as 40 cm.
(20 / x^2) = (-3 / 40^2)
20 * 40^2 = -3 * x^2
x^2 = (20 * 40^2) / (-3)
x^2 = 3200 * 1600 / 3
x^2 ≈ 5.493 * 10^6
x ≈ sqrt(5.493 * 10^6)
x ≈ 2344.28 cm
Therefore, the separation distance between the yellow and green particles is approximately 2344.28 cm.
To determine the separation distance between the yellow particle and green particle, we need to consider the balance of forces acting on the green particle due to the presence of the yellow and red particles.
From the given information, we know that the green particle is at equilibrium. This means that the net force acting on the green particle is zero.
The force between two charged particles can be calculated using Coulomb's law formula:
F = k * (|q1| * |q2|) / r^2
Where:
F is the force between the particles,
k is Coulomb's constant (k = 9 × 10^9 N m^2/C^2),
|q1| and |q2| are the magnitudes of the charges, and
r is the separation distance between the particles.
Since the green particle is at equilibrium, the forces acting on it due to the yellow and red particles must be equal in magnitude but opposite in direction.
First, let's find the force exerted on the green particle by the yellow particle:
F_yellow = k * (|q_green| * |q_yellow|) / r_yellow^2
And let's find the force exerted on the green particle by the red particle:
F_red = k * (|q_green| * |q_red|) / r_red^2
Since the net force on the green particle is zero, we have:
F_yellow = -F_red
Therefore, we can write the equation:
k * (|q_green| * |q_yellow|) / r_yellow^2 = - k * (|q_green| * |q_red|) / r_red^2
The charges of the particles are:
|q_green| = 2.0 C,
|q_yellow| = 5.0 C, and
|q_red| = -3.0 C.
Hence, the equation becomes:
k * (2.0 C * 5.0 C) / r_yellow^2 = - k * (2.0 C * -3.0 C) / r_red^2
Simplifying further:
(2.0 C * 5.0 C) / r_yellow^2 = - (2.0 C * -3.0 C) / r_red^2
(2 * 5) / r_yellow^2 = - (2 * -3) / r_red^2
10 / r_yellow^2 = 6 / r_red^2
Cross multiplying:
10 * r_red^2 = 6 * r_yellow^2
r_red^2 = (6/10) * r_yellow^2
r_red^2 = (3/5) * r_yellow^2
Taking the square root of both sides:
r_red = sqrt((3/5) * r_yellow^2)
Now, we need to find the value of r_yellow.
From the image provided, it appears that the separation distance shown is 40 cm.
So, r_yellow = 40 cm.
Substituting the values into the equation:
r_red = sqrt((3/5) * (40^2) cm)
r_red = sqrt((3/5) * 1600) cm
r_red = sqrt(960) cm
r_red ≈ 31 cm
Therefore, the separation distance between the yellow particle and green particle is approximately 31 centimeters.